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Properties

Label 33.33.2998446588...8481.1
Degree $33$
Signature $[33, 0]$
Discriminant $3^{44}\cdot 11^{60}$
Root discriminant $338.53$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-126686997, -3992061348, -28282285053, 84512905389, 665621580294, -352916325432, -4455495029205, -1695923602263, 9428309081928, 5765703181788, -9369700675533, -6494206038078, 5275486801024, 3841738634745, -1855194763851, -1383513593077, 429771174723, 325326665763, -67410260049, -51815376558, 7225040196, 5687836869, -525348783, -431474868, 25239423, 22354497, -757746, -767778, 12672, 16500, -88, -198, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 198*x^31 - 88*x^30 + 16500*x^29 + 12672*x^28 - 767778*x^27 - 757746*x^26 + 22354497*x^25 + 25239423*x^24 - 431474868*x^23 - 525348783*x^22 + 5687836869*x^21 + 7225040196*x^20 - 51815376558*x^19 - 67410260049*x^18 + 325326665763*x^17 + 429771174723*x^16 - 1383513593077*x^15 - 1855194763851*x^14 + 3841738634745*x^13 + 5275486801024*x^12 - 6494206038078*x^11 - 9369700675533*x^10 + 5765703181788*x^9 + 9428309081928*x^8 - 1695923602263*x^7 - 4455495029205*x^6 - 352916325432*x^5 + 665621580294*x^4 + 84512905389*x^3 - 28282285053*x^2 - 3992061348*x - 126686997)
 
gp: K = bnfinit(x^33 - 198*x^31 - 88*x^30 + 16500*x^29 + 12672*x^28 - 767778*x^27 - 757746*x^26 + 22354497*x^25 + 25239423*x^24 - 431474868*x^23 - 525348783*x^22 + 5687836869*x^21 + 7225040196*x^20 - 51815376558*x^19 - 67410260049*x^18 + 325326665763*x^17 + 429771174723*x^16 - 1383513593077*x^15 - 1855194763851*x^14 + 3841738634745*x^13 + 5275486801024*x^12 - 6494206038078*x^11 - 9369700675533*x^10 + 5765703181788*x^9 + 9428309081928*x^8 - 1695923602263*x^7 - 4455495029205*x^6 - 352916325432*x^5 + 665621580294*x^4 + 84512905389*x^3 - 28282285053*x^2 - 3992061348*x - 126686997, 1)
 

Normalized defining polynomial

\( x^{33} - 198 x^{31} - 88 x^{30} + 16500 x^{29} + 12672 x^{28} - 767778 x^{27} - 757746 x^{26} + 22354497 x^{25} + 25239423 x^{24} - 431474868 x^{23} - 525348783 x^{22} + 5687836869 x^{21} + 7225040196 x^{20} - 51815376558 x^{19} - 67410260049 x^{18} + 325326665763 x^{17} + 429771174723 x^{16} - 1383513593077 x^{15} - 1855194763851 x^{14} + 3841738634745 x^{13} + 5275486801024 x^{12} - 6494206038078 x^{11} - 9369700675533 x^{10} + 5765703181788 x^{9} + 9428309081928 x^{8} - 1695923602263 x^{7} - 4455495029205 x^{6} - 352916325432 x^{5} + 665621580294 x^{4} + 84512905389 x^{3} - 28282285053 x^{2} - 3992061348 x - 126686997 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(299844658869547417502884265471158800440463083305212466036627917931188099334990268481=3^{44}\cdot 11^{60}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $338.53$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1089=3^{2}\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1089}(1024,·)$, $\chi_{1089}(1,·)$, $\chi_{1089}(133,·)$, $\chi_{1089}(265,·)$, $\chi_{1089}(397,·)$, $\chi_{1089}(529,·)$, $\chi_{1089}(661,·)$, $\chi_{1089}(793,·)$, $\chi_{1089}(925,·)$, $\chi_{1089}(1057,·)$, $\chi_{1089}(34,·)$, $\chi_{1089}(166,·)$, $\chi_{1089}(298,·)$, $\chi_{1089}(430,·)$, $\chi_{1089}(562,·)$, $\chi_{1089}(694,·)$, $\chi_{1089}(826,·)$, $\chi_{1089}(958,·)$, $\chi_{1089}(67,·)$, $\chi_{1089}(199,·)$, $\chi_{1089}(331,·)$, $\chi_{1089}(463,·)$, $\chi_{1089}(595,·)$, $\chi_{1089}(727,·)$, $\chi_{1089}(859,·)$, $\chi_{1089}(991,·)$, $\chi_{1089}(100,·)$, $\chi_{1089}(232,·)$, $\chi_{1089}(364,·)$, $\chi_{1089}(496,·)$, $\chi_{1089}(628,·)$, $\chi_{1089}(760,·)$, $\chi_{1089}(892,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{3} a^{8} - \frac{1}{27} a^{7} - \frac{4}{9} a^{6} - \frac{10}{27} a^{5} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{4}{9} a^{7} - \frac{10}{27} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{11} + \frac{2}{9} a^{8} - \frac{2}{27} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{18} + \frac{1}{27} a^{14} + \frac{4}{81} a^{12} + \frac{1}{27} a^{9} + \frac{2}{27} a^{8} - \frac{1}{3} a^{7} - \frac{23}{81} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{7}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{81} a^{19} + \frac{1}{81} a^{13} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{7}{81} a^{7} + \frac{13}{27} a^{5} + \frac{5}{27} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{243} a^{20} - \frac{1}{243} a^{19} + \frac{1}{81} a^{17} + \frac{1}{243} a^{14} + \frac{8}{243} a^{13} - \frac{4}{81} a^{12} - \frac{1}{27} a^{11} - \frac{4}{81} a^{10} - \frac{83}{243} a^{8} - \frac{43}{243} a^{7} - \frac{2}{81} a^{6} - \frac{31}{81} a^{5} + \frac{10}{81} a^{4} - \frac{4}{9} a^{3} - \frac{8}{27} a^{2} - \frac{1}{27} a + \frac{2}{9}$, $\frac{1}{243} a^{21} - \frac{1}{243} a^{19} + \frac{1}{81} a^{17} + \frac{1}{243} a^{15} - \frac{4}{243} a^{13} - \frac{2}{81} a^{12} + \frac{2}{81} a^{11} - \frac{4}{81} a^{10} - \frac{11}{243} a^{9} + \frac{2}{27} a^{8} - \frac{49}{243} a^{7} + \frac{8}{81} a^{6} + \frac{2}{27} a^{5} + \frac{19}{81} a^{4} - \frac{4}{27} a^{3} - \frac{1}{3} a^{2} - \frac{13}{27} a + \frac{4}{9}$, $\frac{1}{243} a^{22} - \frac{1}{243} a^{19} + \frac{1}{81} a^{17} + \frac{1}{243} a^{16} - \frac{4}{81} a^{14} + \frac{2}{243} a^{13} + \frac{1}{27} a^{12} + \frac{2}{81} a^{11} + \frac{4}{243} a^{10} + \frac{1}{27} a^{9} - \frac{23}{81} a^{8} + \frac{62}{243} a^{7} - \frac{1}{9} a^{6} - \frac{13}{27} a^{5} + \frac{34}{81} a^{4} + \frac{13}{27} a^{3} + \frac{2}{9} a^{2} + \frac{11}{27} a + \frac{4}{9}$, $\frac{1}{243} a^{23} - \frac{1}{243} a^{19} + \frac{4}{243} a^{17} - \frac{1}{81} a^{15} - \frac{2}{81} a^{14} - \frac{1}{243} a^{13} + \frac{1}{27} a^{12} + \frac{4}{243} a^{11} - \frac{1}{81} a^{10} - \frac{2}{81} a^{9} - \frac{40}{81} a^{8} + \frac{110}{243} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{4}{81} a^{4} + \frac{1}{27} a^{3} + \frac{1}{3} a^{2} + \frac{11}{27} a + \frac{4}{9}$, $\frac{1}{6561} a^{24} - \frac{1}{729} a^{23} - \frac{2}{2187} a^{22} - \frac{10}{6561} a^{21} + \frac{1}{729} a^{20} - \frac{5}{2187} a^{19} - \frac{2}{6561} a^{18} + \frac{2}{243} a^{17} + \frac{1}{243} a^{16} + \frac{65}{6561} a^{15} - \frac{34}{729} a^{14} + \frac{32}{2187} a^{13} - \frac{98}{6561} a^{12} + \frac{1}{243} a^{11} + \frac{56}{2187} a^{10} - \frac{136}{6561} a^{9} - \frac{152}{729} a^{8} + \frac{56}{243} a^{7} + \frac{134}{729} a^{6} + \frac{11}{243} a^{5} + \frac{38}{81} a^{4} + \frac{37}{81} a^{3} + \frac{28}{81} a^{2} + \frac{14}{81} a + \frac{116}{243}$, $\frac{1}{6561} a^{25} - \frac{2}{2187} a^{23} - \frac{10}{6561} a^{22} + \frac{4}{2187} a^{20} + \frac{25}{6561} a^{19} + \frac{4}{729} a^{18} + \frac{4}{243} a^{17} + \frac{119}{6561} a^{16} + \frac{13}{729} a^{15} + \frac{95}{2187} a^{14} + \frac{361}{6561} a^{13} + \frac{31}{729} a^{12} + \frac{29}{2187} a^{11} + \frac{53}{6561} a^{10} + \frac{2}{81} a^{9} - \frac{25}{81} a^{8} - \frac{1}{729} a^{7} - \frac{115}{243} a^{6} - \frac{8}{81} a^{5} - \frac{38}{81} a^{4} - \frac{29}{81} a^{3} - \frac{1}{81} a^{2} - \frac{46}{243} a + \frac{2}{27}$, $\frac{1}{6561} a^{26} - \frac{10}{6561} a^{23} - \frac{1}{729} a^{22} + \frac{2}{2187} a^{21} - \frac{2}{6561} a^{20} - \frac{1}{243} a^{19} + \frac{5}{2187} a^{18} - \frac{70}{6561} a^{17} + \frac{7}{729} a^{16} + \frac{1}{81} a^{15} + \frac{226}{6561} a^{14} + \frac{35}{729} a^{13} + \frac{103}{2187} a^{12} + \frac{188}{6561} a^{11} - \frac{29}{729} a^{10} - \frac{119}{2187} a^{9} - \frac{364}{729} a^{8} + \frac{2}{81} a^{7} - \frac{44}{243} a^{6} - \frac{13}{81} a^{5} - \frac{8}{27} a^{4} + \frac{2}{81} a^{3} + \frac{80}{243} a^{2} - \frac{11}{27} a - \frac{29}{81}$, $\frac{1}{6561} a^{27} + \frac{1}{729} a^{23} + \frac{2}{2187} a^{21} + \frac{1}{729} a^{20} - \frac{1}{243} a^{19} - \frac{1}{729} a^{18} - \frac{11}{729} a^{17} - \frac{1}{81} a^{16} - \frac{23}{2187} a^{15} + \frac{4}{729} a^{14} - \frac{25}{729} a^{12} + \frac{22}{729} a^{11} - \frac{2}{81} a^{10} + \frac{89}{6561} a^{9} - \frac{104}{729} a^{8} + \frac{1}{243} a^{7} + \frac{260}{729} a^{6} - \frac{64}{243} a^{5} + \frac{8}{27} a^{4} - \frac{16}{243} a^{3} - \frac{38}{81} a^{2} + \frac{2}{27} a - \frac{109}{243}$, $\frac{1}{19683} a^{28} + \frac{1}{19683} a^{27} + \frac{1}{19683} a^{25} + \frac{1}{19683} a^{24} - \frac{11}{6561} a^{23} - \frac{10}{19683} a^{22} + \frac{14}{19683} a^{21} - \frac{8}{6561} a^{20} - \frac{107}{19683} a^{19} + \frac{106}{19683} a^{18} + \frac{4}{2187} a^{17} + \frac{185}{19683} a^{16} + \frac{293}{19683} a^{15} - \frac{4}{6561} a^{14} + \frac{934}{19683} a^{13} + \frac{550}{19683} a^{12} + \frac{122}{6561} a^{11} + \frac{1012}{19683} a^{10} - \frac{731}{19683} a^{9} - \frac{373}{2187} a^{8} - \frac{530}{2187} a^{7} + \frac{199}{2187} a^{6} + \frac{223}{729} a^{5} + \frac{89}{729} a^{4} + \frac{227}{729} a^{3} - \frac{8}{243} a^{2} - \frac{221}{729} a - \frac{209}{729}$, $\frac{1}{19683} a^{29} - \frac{1}{19683} a^{27} + \frac{1}{19683} a^{26} - \frac{1}{19683} a^{24} - \frac{31}{19683} a^{23} - \frac{4}{6561} a^{22} + \frac{37}{19683} a^{21} - \frac{29}{19683} a^{20} - \frac{40}{6561} a^{19} + \frac{107}{19683} a^{18} + \frac{230}{19683} a^{17} - \frac{11}{729} a^{16} + \frac{58}{19683} a^{15} - \frac{404}{19683} a^{14} + \frac{253}{6561} a^{13} + \frac{227}{19683} a^{12} + \frac{808}{19683} a^{11} + \frac{349}{6561} a^{10} - \frac{634}{19683} a^{9} + \frac{38}{2187} a^{8} + \frac{1}{27} a^{7} - \frac{886}{2187} a^{6} - \frac{338}{729} a^{5} - \frac{101}{243} a^{4} - \frac{98}{729} a^{3} - \frac{341}{729} a^{2} + \frac{79}{243} a - \frac{175}{729}$, $\frac{1}{2095865523} a^{30} + \frac{3365}{2095865523} a^{29} - \frac{29200}{2095865523} a^{28} - \frac{26626}{2095865523} a^{27} - \frac{71554}{2095865523} a^{26} - \frac{47293}{2095865523} a^{25} + \frac{2014}{232873947} a^{24} + \frac{4241191}{2095865523} a^{23} + \frac{644410}{2095865523} a^{22} - \frac{548441}{698621841} a^{21} + \frac{749012}{2095865523} a^{20} + \frac{7052072}{2095865523} a^{19} - \frac{173680}{77624649} a^{18} - \frac{38185472}{2095865523} a^{17} + \frac{11473237}{2095865523} a^{16} - \frac{11820473}{698621841} a^{15} - \frac{5950513}{2095865523} a^{14} - \frac{52191556}{2095865523} a^{13} - \frac{66791449}{2095865523} a^{12} - \frac{77064454}{2095865523} a^{11} + \frac{9344468}{2095865523} a^{10} + \frac{30587992}{2095865523} a^{9} - \frac{23200130}{232873947} a^{8} + \frac{72556115}{232873947} a^{7} - \frac{61027757}{232873947} a^{6} + \frac{7518584}{77624649} a^{5} + \frac{14264893}{77624649} a^{4} - \frac{1798729}{25874883} a^{3} - \frac{3117280}{77624649} a^{2} - \frac{34970740}{77624649} a + \frac{4636897}{77624649}$, $\frac{1}{563787825687} a^{31} - \frac{5}{187929275229} a^{30} + \frac{590513}{62643091743} a^{29} + \frac{9864220}{563787825687} a^{28} - \frac{25501078}{563787825687} a^{27} + \frac{12276434}{187929275229} a^{26} + \frac{37841240}{563787825687} a^{25} + \frac{20493479}{563787825687} a^{24} + \frac{122204420}{62643091743} a^{23} - \frac{237218240}{563787825687} a^{22} - \frac{464883848}{563787825687} a^{21} - \frac{97624403}{187929275229} a^{20} - \frac{221396743}{563787825687} a^{19} - \frac{1863675229}{563787825687} a^{18} + \frac{858866005}{187929275229} a^{17} + \frac{9379682374}{563787825687} a^{16} - \frac{2727707417}{563787825687} a^{15} - \frac{401151350}{20881030581} a^{14} - \frac{22399516847}{563787825687} a^{13} + \frac{9238683149}{563787825687} a^{12} - \frac{9532050418}{187929275229} a^{11} + \frac{3901219999}{187929275229} a^{10} + \frac{28534248593}{563787825687} a^{9} + \frac{1587458986}{62643091743} a^{8} - \frac{9678516452}{20881030581} a^{7} - \frac{19331753500}{62643091743} a^{6} + \frac{2247751889}{20881030581} a^{5} - \frac{3693894200}{20881030581} a^{4} + \frac{5873976604}{20881030581} a^{3} + \frac{124604624}{257790501} a^{2} + \frac{2872434358}{6960343527} a + \frac{10051607006}{20881030581}$, $\frac{1}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{32} - \frac{211822085346461810319913728301392945665325240752487789821859784590372611021810846973242602807756733821272924795635}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{31} - \frac{189000889199499419292246389439901272395913011868332370820971641605446761620008181240634970005313481904659447677377990}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{30} + \frac{14044171239425250655836261494151314268087947071926556538332584111181023149952438212511154152787714216396209847252692677995}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{29} + \frac{3427423372595727000128959568574247242653106820645424433997628586536347555006482920883567379968074365766122053541067493617}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{28} + \frac{17809425155440266751542346045084822729112600397565302282490464778467324434798866877183529094361685758215863661867803964273}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{27} + \frac{51565269341616110554269700092556759790289487405381181329061055156340298747088665892320607133121074562059126397100374537549}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{26} - \frac{846467584520271063443206630692417966119217948911874049497111575087299744315028010427535103821533410170684888274898047743}{30372236884380738547325265520957490457548542476458119384798849966189346240245973308035124989291258731957853867008678606775381} a^{25} + \frac{32462233397183980217318225545953664298709045710521300301614386424399242101723884347229475297474672577040333541979644304019}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{24} - \frac{1445419073660658158062875641292164679357499428256921041448219791850604946233349256410405517333526644859300260145630011015277}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{23} + \frac{101325812524665418916516717085164509169016511887980395007443263927435729540936961651072263508656933495621633498882482719939}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{22} + \frac{305105934581100527516127563630574642400882326080846833724525154604069373727337595706285511730938583614142327261770501939238}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{21} + \frac{1521720624228445986445044308624980757072187696072535654799164012922057451827495557730212940633838323756070241982032886187680}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{20} - \frac{245482833928964851444244309791640002135383101735490380554357996440748470392199291157266528758061123765550664190112619275169}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{19} - \frac{3999328327803575200006936588453780591231959891889864024769549017791660412290195872154241648010113868515474642081569329767079}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{18} + \frac{4263609255436131751609294563239210461697811283983196217874921950986631360947284883120875693387022249132243105758373130839300}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{17} + \frac{1190584036418152114067481608839098763078627934208797334157030278776057986509673968210812165631657872969310056734182988562926}{91116710653142215641975796562872471372645627429374358154396549898568038720737919924105374967873776195873561601026035820326143} a^{16} - \frac{3432761442740412866579539799537394333901871487128934242901138410687726041573222919564194745365424807617423376505776494269190}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{15} + \frac{7176495365835921182460881254399251018359078811654581879261458281403254555077909765537730752473681091267299782084373695381856}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{14} + \frac{8101992897440594885561862800786713010766897153059672579592491203521126556362847207272226417559275104428568634754674806311272}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{13} - \frac{4208155576836073159818580361999773867613846751042247682981378508649176698856225843666479042002177272083060197461077070794905}{91116710653142215641975796562872471372645627429374358154396549898568038720737919924105374967873776195873561601026035820326143} a^{12} + \frac{11593449288224165200807789618605253823518083595932452461902513574236571002612131333788960768692866851246116074287584707742781}{273350131959426646925927389688617414117936882288123074463189649695704116162213759772316124903621328587620684803078107460978429} a^{11} - \frac{1490728766747050544673765700719636994012509519810837219119406550157352421880883810810780968874825165247282310578837395735970}{30372236884380738547325265520957490457548542476458119384798849966189346240245973308035124989291258731957853867008678606775381} a^{10} + \frac{9103934797983100096205234674290651007696890338939752112962036851720959696263202026600143692712924161193894302294099808654947}{820050395878279940777782169065852242353810646864369223389568949087112348486641279316948374710863985762862054409234322382935287} a^{9} - \frac{26872572382500947913581042998232469275233198142199897155552953832208474855234418557985992322634385273672874706367335042398334}{91116710653142215641975796562872471372645627429374358154396549898568038720737919924105374967873776195873561601026035820326143} a^{8} + \frac{4934441527878973152337382522441088919366014944928645029408389200836506107851680924266976657421599493411478645185429256300745}{30372236884380738547325265520957490457548542476458119384798849966189346240245973308035124989291258731957853867008678606775381} a^{7} - \frac{27728433390977616257551506972014031686628091583789482979784867121194160368884163966971525348804081933753599970468223324800083}{91116710653142215641975796562872471372645627429374358154396549898568038720737919924105374967873776195873561601026035820326143} a^{6} + \frac{1737551468831341232898990914033221171115809101440944256293035469471206612785365643088045421619157578132053976086586128499758}{10124078961460246182441755173652496819182847492152706461599616655396448746748657769345041663097086243985951289002892868925127} a^{5} + \frac{3759287269394695673536174924075466618116415663136452516101220910128448415750570298048225532199602080236175975465332474823449}{10124078961460246182441755173652496819182847492152706461599616655396448746748657769345041663097086243985951289002892868925127} a^{4} - \frac{2400945483563263933404262390733256991485347235696142897752400295923237016717025480798789749579947705050939531363641210233642}{30372236884380738547325265520957490457548542476458119384798849966189346240245973308035124989291258731957853867008678606775381} a^{3} - \frac{2276854075132115121081549438299287401486077973905671203511523110240041844971008909660363446961678689900847771985905402339020}{10124078961460246182441755173652496819182847492152706461599616655396448746748657769345041663097086243985951289002892868925127} a^{2} + \frac{711824207102893532297600980971540153236539310914731198063753695406548684089951299793655214743617231062400396720416321878033}{3374692987153415394147251724550832273060949164050902153866538885132149582249552589781680554365695414661983763000964289641709} a - \frac{6602612723386511249620952760607339393819982610534348586475957462440356607514351271594931530641965989592448098115079774518779}{30372236884380738547325265520957490457548542476458119384798849966189346240245973308035124989291258731957853867008678606775381}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $32$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1383473472152591500000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 11.11.672749994932560009201.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $33$ R $33$ $33$ R $33$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
11Data not computed